NONTRIVIAL SOLUTIONS OF A DIRICHLET BOUNDARY VALUE PROBLEM WITH IMPULSIVE EFFECTS

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1 Dynamic Systems and Applications NONRIVIAL SOLUIONS OF A DIRICHLE BOUNDARY VALUE PROBLEM WIH IMPULSIVE EFFECS JOHN R. GRAEF, SHAPOUR HEIDARKHANI, AND LINGJU KONG Department of Mathematics, University of ennessee at Chattanooga, Chattanooga, N 3743, USA Department of Mathematics, Faculty of Sciences, Razi University Kermanshah 67149, Iran Department of Mathematics, University of ennessee at Chattanooga, Chattanooga, N 3743, USA ABSRAC. By using variational methods and critical point theory, the authors establish criteria for the existence of nontrivial classical solutions to a class of Dirichlet boundary value problems with impulsive effects. he results are illustrated with two examples. AMS MOS Subject Classification. 34B37, 34B15, 47J1. 1. INRODUCION We consider the nonlinear Dirichlet boundary value problem consisting of u x = λfx, ux + gux, a.e. x [, ], 1.1 u = u =, together with the impulsive conditions 1. u x j = I j ux j, j = 1,..., p, where λ is a positive parameter, >, f : [, ] R R is an L 1 -Carathéodory function, x = < x 1 < < x p < x p+1 =, and u x j is defined by u x j = u x + j u x j = lim x x + j u x lim x x j u x. hroughout this paper, we assume the following conditions hold without further mention: H1 g : R R is a Lipschitz continuous function with a Lipschitz constant L, 4/, i.e., gt 1 gt L t 1 t for all t 1, t R, and g = ; H he impulsive functions I j : R R, j = 1,...,p, are continuous and satisfy the condition p I jt 1 I j t t 1 t for all t 1, t R; Received July, $15. c Dynamic Publishers, Inc.

2 336 J. R. GRAEF, S. HEIDARKHANI, AND L. KONG H3 I j, j = 1,..., p, have sublinear growth, i.e., there exist constants a j >, b j, and γ j [, 1 such that I j t a j + b j t γ j for every t R and j = 1,...,p. he impulsive problem 1.1, 1. will be referred to as IP in the remainder of this paper. By employing a consequence of a local minimum theorem due to Bonanno see Lemma.1 below, we establish the existence of nontrivial classical solutions to the problem IP. Under suitable algebraic assumptions on the nonlinear term f, we first prove the existence of at least one nontrivial classical solution, and then we obtain the existence of a second classical solution by further assuming that ft, for all t [, ] and asking that the following condition of Ambrosetti-Rabinowitz AR type holds: A1 there exist s > 4+L 4 L and R > such that < sfx, t tfx, t for all t R and x [, ]. he role of this condition is to ensure the boundedness of the Palais-Smale sequences for the Euler-Lagrange functional associated with the problem IP. his is important in applications of critical point theory. he purpose of the condition ft, for all t [, ] is to avoid the existence of the trivial solution to the problem IP. Impulsive differential equations arise from real world processes that describe the dynamics in which sudden and discontinuous jumps occur. Because of the importance of such problems, they have been studied by many authors; for example, see [, 14, 1, 3] and the references therein. In particular, the existence and multiplicity of solutions for impulsive differential equations have been examined in many works, and for an overview on this subject, we refer the reader to the monographs [6, 1] and the papers [1, 3, 4, 5,, 1, 16, 15, 17,, 4, 6, 7, ]. We especially refer the reader to the paper [9] in which, using Lemma.1 below, existence results for two and three nontrivial solutions are established for a class of Sturm-Liouville problems with mixed conditions and the p-laplacian.. PRELIMINARIES Our main tool, Lemma.1 below, is a consequence of an existence result for a local minimum [7, heorem 5.1] that in turn was inspired by Ricceri s variational principle []. In it, an inequality on the functionals Φ and Ψ is needed. For a given nonempty set X and two functionals Φ, Ψ : X R, we define ϑr 1, r = inf v Φ 1 r 1,r sup u Φ 1 r 1,r Ψu Ψv r Φv

3 NONRIVIAL SOLUIONS 337 and Ψv sup u Φ ρr 1, r = sup 1,r 1 Ψu v Φ 1 r 1,r Φv r 1 for all r 1, r R with r 1 < r. In what follows, we let X denote the dual space of X. Lemma.1 [7, heorem 5.1]. Let X be a real Banach space, Φ : X R be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on X, and let Ψ : X R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Let I λ = Φ λψ and assume that there exist r 1, r R with r 1 < r such that hen, for each λ 1 ρr 1,r, 1 ϑr 1,r I λ u for all u Φ 1 r 1, r and I λ u,λ =. product ϑr 1, r < ρr 1, r., there exists u,λ Φ 1 r 1, r such that I λ u,λ Let X = H 1, and H, = {u C 1 [, ] : u L [, ]}. he inner in X induces the norm u, v = u = u xv xdx, 1 u x dx. Definition.. By a classical solution of the problem IP, we mean a function u {ux H 1, : ux H x j, x j+1, j =, 1,..., p} such that u satisfies 1.1 and 1.. We say that a function u X is a weak solution of the problem IP if u xv xdx+ for every v X. I j ux j vx j guxvxdx λ fx, uxvxdx = Remark.3. Using standard methods, it is easy to verify that a weak solution of IP is also a classical solution e.g., see [3, Lemma 5]. Definition.4. Let X be a real reflexive Banach space. If any sequence {u n } X, such that {Ju n } is bounded and J u n as n, possesses a convergent subsequence, then we say that J satisfies the Palais-Smale PS condition. Here, the sequence {u n } is called a PS sequence. Next, we let Fx, t = t fx, ξdξ for all x, t [, ] R,

4 33 J. R. GRAEF, S. HEIDARKHANI, AND L. KONG t Gt = and define the functional J : X R by Ju = 1 u x dx + uxj gξdξ for all t R, I j tdt + Guxdx λ Fx, uxdx. From the continuity of f, g, and I j, we see that Ju is strongly continuous in X, J C 1 X, and the Gâteaux derivative of J is J u, v = u xv xdx + I j ux j vx j guxvxdx λ fx, uxvxdx for every u, v X. We need the following lemmas to prove our main result. Lemma.5. Assume that A1 holds. hen, J satisfies the PS condition. Proof. Let {u n } be a sequence in X such that lim n J u n = and Ju n is bounded. From H3, we see that.1 I j u n x j u n x j a j u n x j + b j u n x j γ j+1. hen, in view of the fact that. u = max t [,] ut we have u for u X,.3 I j u n x j u n x j a j u n x j + b j u n x j γ j+1 a j u n + b j γj +1 u n γ j+1. Again, by H3, we have uxj I j xdx a j ux j + b j γ j + 1 ux j γj+1.

5 hus, from., it follows that uxj I j xdx.4 NONRIVIAL SOLUIONS 339 uxj I j xdx a j ux j + b j γ j + 1 ux j γ j+1 a j u + b j γ j + 1 γj +1 u γ j+1. From A1,.3, and.4, we see that there exists C > such that s sju n J u n, u n = 1 u n + λ [fx, u n xu n x sfx, u n x]dx + + s [sgu n x + gu n xu n x]dx uxj I j xdx I j u n x j u n x j s 1 u n 1 s 4 L + 1 u n s a j u n + b γj +1 j u n γ j+1 γ j + 1 γj +1 a j u n + b j u n γ j+1 C [ s = 1 L 1 + L ] u n 4 4 s a j u n + b γj +1 j u n γ j+1 γ j + 1 γj +1 a j u n + b j u n γ j+1 C. his implies that {u n } is bounded in X. Next, we show that {u n } converges strongly to some u in X. Since {u n } is bounded in X, there exists a subsequence of {u n } denoted again by {u n } such that {u n } converges weakly to some u in X. hen, {u n } converges uniformly to u on [, ]. hus, u n u, u n u, I j u n x j I j ux j u n x j ux j,

6 34 J. R. GRAEF, S. HEIDARKHANI, AND L. KONG and gu n x guxu n x uxdx, fx, u n x fx, uxu n x uxdx, as n. Since lim n + J u n = and {u n } converges weakly to u X, we have Note that J u n J u, u n u = J u n J u, u n u as n. + λ u nx u x dx I j u n x j I j ux j u n x j ux j gu n x guxu n x uxdx fx, u n x fx, uxu n x uxdx. hen, we have u n u as n. hus, {u n } converges weakly to u in X, and so the functional J satisfies the PS condition. he following lemma can be found in [11, Lemma.]. Lemma.6. Let : X X be the operator defined by.5 uv = u xv xdx + I j ux j vx j guxvxdx for every u, v X. hen admits a continuous inverse on X. We now recall the following classic mountain pass lemma of Ambrosetti and Rabinowitz see, for example, [13, heorem 7.1]. Below, we denote by B r u the open ball centered at u X with radius r >, B r u its closure, and B r u its boundary. Lemma.7. Let X, be a real Banach space and I C 1 X, R. Assume that I satisfies the PS condition and there exist ū, û X and ρ > such that I1 ū B ρ û; I max{iū, Iû} < inf u Bρû Iu. hen, I possesses a critical value which can be characterized as where c = inf max Iγs inf Iu, γ Γ s [,1] u B ρu Γ = {γ C[, 1], X : γ = u, γ1 = u 1 }.

7 and NONRIVIAL SOLUIONS MAIN RESULS We begin by defining the constants see [3] C 1 = 1 b j γ j + 1 C 3 = C = 1 + a j + b j γ j + 1 b j γ j + 1 γj +1 γj +1,, γj +1 and the functions H 1, H : [, R by 3.1 H 1 t = C 1 L t C 3 t and H t = C + L t + C 3 t. For u X, define the functionals Φ, Ψ : X R by 3. Φu = 1 uxj u x dx + I j tdt + and 3.3 Ψu = Fx, uxdx., Guxdx In the sequel, given a non-negative constant ν 1 and four positive constants ν, τ, η, and δ with η, δ < /, 4 C 1 L ν 1 C 3 ν 1 η + δ ηδ C + L τ η + δ + ηδ C 3τ, and 4 C + L ν 3 + C 3 ν 3 η + δ ηδ C + L τ η + δ + ηδ C 3τ, we let and a τ ν 1 = b τ ν = 4 4 C1 L C + L sup t [ ν 1,ν 1 ] Fx, tdx δ η ν 1 C 3 ν 1 η + δ ηδ C + L sup t [ ν 3,ν 3 ] Fx, tdx δ η ν + C 3 ν η + δ ηδ C + L Fx, τdx τ η+δ ηδ C 3τ Fx, τdx τ η+δ ηδ C 3τ,

8 34 J. R. GRAEF, S. HEIDARKHANI, AND L. KONG provided ν 3 > satisfies 3.4 H ν 3 = sup H u u Φ,r ] and 3.5 r = H ν = 4 C + L ν + C 3 ν. Remark 3.1. It is easy to see that if u Φ, r ], then u < B <. hus, 3.4 is well defined. Moreover, it is easy to check that ν 3 ν. We now present our main existence result. heorem 3.. Assume that C 1 L > and there exist five positive constants ν 1, ν, τ, η, and δ with η, δ < /, τ > ν 1 > C 3 η+δ, and τ < ηδ ν such that C 1 L A Fx, t for all x, t [, η] [ δ, ] [, τ]; A3 b τ ν < a τ ν 1. hen, for each λ Λ 1 = 1 a τν 1, 1 b τν classical solution u 1 X such that 3.6 r 1 < 1 where u 1 x dx r 1 = 4, the problem IP has at least one nontrivial u1 x j C 1 L I j tdt + ν 1 C 3 ν 1. Gu 1 xdx < r, If we further assume that ft, for all t [, ] and A1 holds, then for each λ Λ 1, the problem IP has at least two distinct nontrivial classical solutions u 1, u X with u 1 satisfying 3.6. Proof. Let the functionals Φ and Ψ be given in 3. and 3.3, respectively. It is well known that Ψ is a Gâteaux differentiable functional, is sequentially weakly lower semicontinuous, and its Gâteaux derivative at u X is the functional Ψ u X defined by Ψ uv = fx, uxvxdx for every v X. o show that Ψ : X X is a compact operator, let u n u X weakly in X as n. hen, u n u strongly in C[, ]. Since fx, is continuous in R for every x [, ], we have fx, u n fx, u strongly as n. By the Lebesgue dominated convergence theorem, we see that Ψ u n Ψ u strongly. hus, Ψ is strongly continuous on X, which implies that Ψ is a compact operator by [5, Proposition 6.].

9 NONRIVIAL SOLUIONS 343 We also know that Φ is a Gâteaux differentiable functional whose Gâteaux derivative at u X is the functional Φ u X given by Φ uv = u xv xdx+ I j ux j vx j guxvxdx for every v X. Moreover, Lemma.6 implies that Φ admits a continuous inverse on X. Since Φ is monotonic, Φ is sequentially weakly lower semicontinuous see [5, Proposition 5.]. Since u γ j+1 u for every u 1 and u γ j+1 u for every u < 1, we have that u γ j+1 u + u for all u X. Condition H1 implies 3. gt L t for all t R, Hence, from.,.4, and 3., we see that 3.9 C 1 L u C 3 u Φu C + L u + C 3 u and Φ is coercive. Let τ x, η x [, η], 3.1 wx = τ, x [η, δ], τ x, x [ δ, ]. δ It is clear that w X and 3.11 w = η + δ ηδ τ. Since C 1 L > and C 3 >, we have H 1 t for t [ [ C is strictly increasing on 3 C 1 L,, and H 1 t on δ, / implies > 4. Since τ > ν η δ 1 > C 3 η+δ ν ηδ 1 > ν 1. Moreover, ν 1 > C3 3.11, we see that w > ν 1 > C 3 [ is strictly increasing on C 3, C 1 L C 1 L C 1 L, implies H 1 w > H 1 ν 1 > H 1 C 1 L [, C 3 C 1 L C 3, C 1 L >, we have ], H 1 t. Now η, η+δ τ > ηδ implies ν 1 > C 3. Hence, from C 1 L, which, together with the fact that H 1 t C 3 C 1 L =.

10 344 J. R. GRAEF, S. HEIDARKHANI, AND L. KONG hus, r 1 = H 1 ν 1, and so 3.1 C 1 L w C 3 w > r 1 >. [ With H and r defined by 3.1 and 3.5, we see that H t is strictly increasing C on 3 η+δ,. Since < τ < η+δ ηδ ν, we have H τ < r ηδ = C 1 L H ν, i.e., 3.13 C + L Hence, from 3.9, 3.1, and 3.13, we obtain w + C 3 w < r. r 1 < Φw < r. [ C Since H 1 t is strictly increasing on the interval 3,, C 1 L ν 1 > C 3, and C 1 L r 1 = H 1 ν 1, we see that [ ] C , ν 1 {t [, : H 1 t r 1 } C 1 L and 3.15 Recalling that H 1 t for t 3.16 ν 1, {t [, : H 1 t r 1 } =. [, From , we have C 3 C 1 L [, ] C 3 C 1 L ], we have {t [, : H 1 t r 1 } {t [, : H 1 t r 1 } = [, For any u Φ 1, r 1 ], from 3.9, we observe that H 1 u Φu r 1. ] ν 1. Hence, using 3.17, we have u ν 1. hen, in view of., we see that Φ 1, r 1 ] {u X : u ν 1 }. hus, sup Ψu = u Φ 1,r 1 ] sup Fx, uxdx u Φ 1,r 1 ] sup Fx, tdx. t [ ν 1,ν 1 ]

11 NONRIVIAL SOLUIONS 345 Since wx τ for each x [, ], condition A implies 3.1 herefore, ρr 1, r η Fx, u 1 xdx + δ Ψw sup u Φ 1,r 1 ] Ψu Φw r 1 Ψw sup t [ ν 1,ν 1 ] Fx, tdx Φw r 1 δ Fx, u 1 xdx. Fx, τdx η sup t [ ν 1,ν 1 ] Fx, tdx η + δ ηδ C + L τ η+δ + C ηδ 3τ 4 C 1 L ν 1 + C 3 ν 1 = a τ ν 1. On the other hand, if we let r 3 = sup u Φ,r ] H u, then, for any u Φ, r ], H u r 3. his, together with 3.4 and the fact that H is increasing on [,, implies that u ν 3. hus, from., it follows that hen, hus, Φ 1, r ] {u X : u ν 3 }. sup Ψu = u Φ 1,r ] sup Fx, uxdx u Φ 1,r ] sup Fx, tdx. t [ ν 3,ν 3 ] ϑr 1, r sup u Φ 1,r Ψu Ψu 1 r Φu 1 sup t [ ν 3,ν 3 ] Fx, tdx Ψu 1 r Φu 1 sup t [ ν 3,ν 3 ] Fx, tdx δ Fx, τdx η 4 C 1 L ν + C 3 ν η + δ ηδ C + L τ η+δ C ηδ 3τ = b τ ν. Hence, from A3, we have ϑr 1, r b τ ν < a τ ν 1 ρr 1, r. herefore, applying Lemma.1, we obtain that for each λ Λ 1, the functional J = Φ λψ has at least one critical point u 1 X that is a local minimum of J such that r 1 < Φu 1 < r. hat is, 3.6 holds. Now, we establish the existence of a second local minimum of J distinct from u 1. Without loss of generality, we may assume that u 1 is a strict local minimum for

12 346 J. R. GRAEF, S. HEIDARKHANI, AND L. KONG the functional J in X. hen, there exists ρ > such that inf u u1 =ρ Ju > Ju 1. Choose ũ X \ {}. From A1, there exist a, b > such that Jtũ = 1 tũxj tũ x dx + I j tdt + Gtũxdx λ t ũ x dx + tũxj I j tdt + Lt λt s a ũx s dx + b as t. ũx dx Fx, tũxdx hus, there exists t large enough so that J tũ < inf u u1 =ρ Ju. Hence, in view of Lemma.5, all the conditions of Lemma.7 are satisfied with û = u 1 and ū = tũ, so there is a critical point u of J such that Ju > Ju 1. Clearly, the condition that ft, for all t [, ] implies that u is nontrivial. Finally, taking into account Remark.3 and the fact that the weak solutions of the problem IP are exactly the critical points of the functional J, completes the proof of the theorem. Remark 3.3. From the proof of heorem 3. see 3.9, we have H 1 u Φu H u for u X. We now apply heorem 3. to the case where the function f in IP is separable. Let α L 1 [, ] be such that αx a.e. x [, ], α, and let h : R R be a continuous function such that ht on [, ] and h. Set Ht = t hξdξ for all t R. Corollary 3.4. Assume that C 1 L > and there exist five positive constants ν 1, ν, τ, η, and δ with η, δ < /, τ > ν 1 > C 3 η+δ, and τ < ηδ ν such that: A4 b τ ν < a τ ν 1, where a τ ν 1 = 4 C 1 L α L 1 [,]Hν 1 α L 1 [η, δ]hτ C + L C1 L ν 1 C 3 ν 1 η + δ ηδ τ η+δ ηδ C 3τ α L b τ ν = 1 [,]Hν 3 α L 1 [η, δ]hτ 4 C + L ν + C 3 ν η + δ, η+δ C + L τ ηδ ηδ C 3τ and ν 3 satisfies hen, for each λ Λ 3 =, 1 a τν 1 b τν, the problem u x = λαxhux + gux, a.e. x [, ], 3.19 u x j = I j ux j, j = 1,,..., p, u = u =,,

13 NONRIVIAL SOLUIONS 347 has at least one nontrivial classical solution u 1 X satisfying 3.6. If we further assume that A5 there exist s > 4+L 4 L and R > such that < sht tht for all t R, then, for each λ Λ 3, the problem 3.19 has at least two distinct nontrivial classical solutions u 1, u X with u 1 satisfying 3.6. We conclude this paper with two examples to illustrate our results. Example 3.5. Consider the problem u x = λ 3 u1/ u, a.e. x [, 4], 3. u = u4 =, u x 1 = 1, < x 1 < 4. We claim that there exists λ > such that, for each λ > λ, the problem 3. has at least one nontrivial classical solution. In fact, with = 4, p = 1, αx = 1, hu = 3 u1/ +1, gu = 1 u, and I 1u = 1, we see that problem 3. is of the form of the problem 3.19 and the covering assumptions H1 H3 are satisfied. In particular, in H1, we can take L = 1, and in H3, we can choose a 1 = 1 and b 1 = γ 1 =. From the definition of C 1, C, and C 3, we have C 1 = C = 1 and C 3 = 1. Clearly, C 1 L = 1 4 >. Choose η = δ = 1 and ν 1 > C 3 a τ ν 1 = C 1 L 4 ν 3/ 1 + ν 1 = 4. Note that τ 3/ + τ 1 4 ν 1 ν 1 3 τ τ hen, there exists τ > ν 1 large enough so that a τ ν 1 >. For large ν >, let r be defined by 3.5. hen, for u Φ 1, r ], by 3.1 and Remark 3.3, we have hus, H 1 u = 1 4 u u φu r. 3.1 u r. Again, by 3.1 and Remark 3.3, we see that 3. H u φu H u H 1 u = 1 4 u + u..

14 34 J. R. GRAEF, S. HEIDARKHANI, AND L. KONG In the remainder of this example, let D i, i = 1,...,6, denote some appropriate constants. For u Φ 1, r ], from 3.1 and 3., it follows that H u r r r 4 D 1 r + D. hen, in view of 3.4 and 3.5, we have ν3 3 H ν 3 = H D 1 r + D = D 1 4 ν + ν i.e., 3 4 ν 3 + ν 3 D 3 ν + D 4. his, together with Remark 3.1, implies that 3.3 ν ν 3 D 5 ν + D 6. Hence, 4 ν 3/ + ν τ 3/ + τ + D D 3 ν + D 4, 3 4 ν + ν 3 τ τ 4 ν 3/ 3 + ν 3 τ 3/ + τ b τ ν = 3 4 ν + ν 3 τ τ 4 D 5 ν + D 6 3/ + D 5 ν + D 6 τ 3/ + τ 3 4 ν + ν 3 τ τ hus b τ ν for ν large enough and b τ ν as ν. Note that 1 b τν as ν. hen, the claim follows from the first part of Corollary 3.4. Remark 3.6. In Example 3.5, since a τ ν 1 > and b τ ν as ν, then we η+δ can fix a ν large enough so that τ < ηδ ν and b τ ν < a τ ν 1. Example 3.7. Let ν be fixed as in Remark 3.6. hen, b τ ν < a τ ν 1. Choose ζ > large enough so that ζ > D 5 ν + D 6, where D 5 and D 6 are the constants given in Example 3.5. Let h : R R be continuous satisfying 3 ht = t1/ + 1, t ζ 3t, t > ζ. Consider the problem u x = λhu + 1 u, a.e. x [, 4], 3.4 u = u4 =, u x 1 = 1, < x 1 < 4..

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